Sending, receiving, and organizing email in Outlook.com
Can you get into your account at all right now? If so, go to https://account.live.com > Security > Manage how I sign in, add the new email before you delete the old one.
If you cannot get in because you cannot get to the code, look for an option like “Show more verification methods” to see if you have a backup (e.g., a phone number or alternate email).
If you don’t have access to any of the options listed, select “I don’t have any of these.” This will begin the recovery process. As part of that process, Microsoft will let you add new contact information — but only after a 30-day waiting period.
This 30-day delay is intentional. It’s a security safeguard designed to protect users in case someone else tries to hijack their account by changing recovery info. The wait gives the rightful owner time to notice and stop unauthorized changes.
Unfortunately, this policy cannot be bypassed. Microsoft explains it further here.